Nonexistence of spatially localized free vibrations for a class of nonlinear wave equations
type: Item
instance of: scholarly article
title: Nonexistence of spatially localized free vibrations for a class of nonlinear wave equations
zbMATH Open document ID: 0656.35093
DOI: 10.1007/BF02564463
published in: Commentarii Mathematici Helvetici
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140099
review text: We prove the nonexistence of free vibrations of arbitrary period with polynomially decreasing profiles for a large class of nonlinear wave equations in one space dimension. Our class of admissible models includes examples of non integrable wave equations with certain polynomial nonlinearities, as well as examples of completely integrable ones with exponential nonlinearities related to Mikhailov's equations. Our result thus proves a particular case of a conjecture first formulated by Eleonskii, Kulagin, Novozhilova and Silin, and dispels some confusion regarding the relationship between the existence of so-called breather- solutions and the complete integrability of the wave equation. Our class of admissible nonlinearities also contains a particular instance of the nonlinear scalar Higgs' equation, but does not contain the Sine-Gordon equation which is known to possess a \(2\pi\)-periodic solution in time with exponential fall-off in the spatial direction. Our results may be considered as complementary to recent results by Coron and Weinstein. Our arguments are entirely global, and rest upon methods from the calculus of variations.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4071501
zbMATH Keywords: nonexistence
zbMATH Keywords: free vibrations
zbMATH Keywords: polynomially decreasing
zbMATH Keywords: nonlinear wave equations
zbMATH Keywords: non integrable
zbMATH Keywords: completely integrable
zbMATH Keywords: Sine-Gordon equation
author: Pierre-A. Vuillermot
MaRDI profile type: Publication
OpenAlex ID: W2086963993
description: scientific article; zbMATH DE number 4071501